Ratio of circumscribed triangle to inscribed triangle

• Mar 20th 2013, 12:45 PM
Ratio of circumscribed triangle to inscribed triangle
Equilateral triangle ABC has a circle inscribed in it. Inside the circle is inscribed another equilateral triangle, DEF. What is the ratio of AB to DE? (Answer from back of book: 4 to 1). I would expect the answer to be 2, but we don't know how to get the answer.

I tried make a picture of what I mean (attached), but the letters came out wrong.

Edit: Ignore the point labels in the figure. I need the ratio of one side of the big triangle to one side of the little triangle. Thanks.
• Mar 20th 2013, 12:59 PM
MINOANMAN
Re: Ratio of circumscribed triangle to inscribed triangle

your figure is not correct. you put D at the barycenter of the triangle....

if you are searching for a ratio between the initial triangle ABC and the one inscribed then the ratio of their areas is 4:1
• Mar 20th 2013, 01:30 PM
Re: Ratio of circumscribed triangle to inscribed triangle
I know, but I am not skilled enough with Geogebra to change the letters. I said that in my first post, but thought I included enough information to make it clear what I wanted.

I need the ratio of one side of the big triangle to one side of the small triangle. Thanks.
• Mar 20th 2013, 01:49 PM
Plato
Re: Ratio of circumscribed triangle to inscribed triangle
Quote:

I know, but I am not skilled enough with Geogebra to change the letters. I said that in my first post, but thought I included enough information to make it clear what I wanted.
I need the ratio of one side of the big triangle to one side of the small triangle. Thanks.

$\displaystyle m(\overline{EH})=\frac{m(\overline{AB})}{2}$.

$\displaystyle m(\overline{ED})=\frac{m(\overline{AE})}{3}$

$\displaystyle m(\overline{AE})=\frac{\sqrt{3}m(\overline{AB})}{2 }$
• Mar 20th 2013, 04:49 PM
johng
Re: Ratio of circumscribed triangle to inscribed triangle

Attachment 27619
• Mar 20th 2013, 05:04 PM
Plato
Re: Ratio of circumscribed triangle to inscribed triangle
Quote:

Originally Posted by johng
Attachment 27619

There is a problem with you diagram.
It is not labeled the same as the one on the OP.

He did ask about $\displaystyle \overline{ED}$.
Now, I will grant you that there seems to be a contradiction in his post.
• Mar 20th 2013, 05:06 PM
Re: Ratio of circumscribed triangle to inscribed triangle
Quote:

Originally Posted by johng

Yes, very nice. Just the way I thought (the book must have a typo).
• Mar 20th 2013, 05:11 PM
He did ask about $\displaystyle \overline{ED}$.